Radar system and method for detecting and tracking a target

ABSTRACT

A radar system for detecting and tracking at least one target utilizing a mechanically rotated two-dimensional radar antenna system with a fan-shaped beam arrangeable on a non-stable radar platform. The radar system includes a tracking filter configured to estimate an azimuth angle of the at least one target with respect to a fixed reference coordinate system based on: azimuth angle information of at least one target radar return signal measured utilizing the radar antenna system with respect to a local coordinate system of the radar platform, and radar platform relative orientation with respect to the fixed reference coordinate system at the time of the at least one target radar return signal, such that a software-based motion-compensation of the radar platform is provided.

TECHNICAL FIELD

The present invention relates to the field of 2D search radar systems,especially for use in the maritime and aeronautical applications whereweight and costs of the radar system is of importance, but also otherapplications where motion compensation of a radar antenna is requiredmight be of interest.

BACKGROUND ART

One common radar type is a radar system that can provide information ofa detected target's azimuth and range. This type of radar system isoften called a bidimensional (2D) radar system. By mechanically rotatingthe radar antenna around an axis which is orthogonal to the horizontalplane, a 2D radar system can effectively cover a 360° angle area. Toadequately detect targets at different elevations, a radar antennagenerating a vertical fan beam is used, i.e. a beam narrow on theazimuth plane and tall in the elevation plane. This type of radarsystems are commonly used in navigation and air warning radarapplications.

When a radar antenna of 2D radar system experiences roll and pitchmotion, for example when arranged on a marine vessel, said radar systemhas problems in accurately tracking detected targets because of thevarying divergence between the radar antenna's rotational axis and theorthogonal of the horizontal plane, i.e. the difference between avarying radar system's local coordinate system and a static horizontalcoordinate systems.

The solution to this problem has been to arrange to the radar antenna ona servo based motion compensating support, which compensates roll andpitch motion of the radar antenna with respect to a horizontalcoordinate system by means of inertial sensors, a control system and aservo system that stabilizes the orientation of the radar antenna, suchthat the rotating axis of the radar antenna is always orthogonal to thehorizontal plane. Such a solution is for example known from patentdocument JP2006311187A. The present servo systems are however expensive,heavy and a potential source of unreliability.

Another disadvantage using a 2D search radar system having a verticalfan beam antenna is that it cannot provide information about targetelevation, and the target data is thus limited to azimuth, range andradial velocity. When elevation information is needed, an additionalheight-finding radar antenna must be provided, or a different type ofradar system must be used, for example phased array radar systems.

There is thus a need for an improved 2D radar system, which partlyavoids the above mentioned disadvantages.

SUMMARY

The object of the present invention is to provide a radar system fordetecting and tracking at least one target by means of a mechanicallyrotated two-dimensional (2D)-radar antenna system with a fan-shapedbeam, arrangeable on a non-stable radar platform where the previouslymentioned problems are partly avoided. This object is achieved by thecharacterizing portion of claim 1, where said radar system comprises atracking filter configured to estimate an azimuth angle of said at leastone target with respect to a fixed reference coordinate system,preferably a fixed horizontal coordinate system, based on:

-   -   azimuth angle information of at least one target radar return        signal measured by means of said radar antenna system with        respect to a local coordinate system of said radar platform,    -   radar platform relative orientation with respect to said fixed        reference coordinate system at the time of said at least one        target radar return signal,    -   such that a software-based motion-compensation of said radar        platform is provided.

The object of the present invention is also to provide a method fordetecting and tracking at least one target by means of a mechanicallyrotated two-dimensional (2D) radar antenna system with a fan-shapedbeam, arrangeable on a platform where the previously mentioned problemsare partly avoided. This object is achieved by the characterizingportion of claim 6, wherein said method comprises the following steps:

-   -   obtaining azimuth angle information of at least one target radar        return signal measured by means of said radar antenna system        with respect to a local coordinate system of said radar        platform,    -   obtaining radar platform relative orientation with respect to a        fixed reference coordinate system at the time of said at least        one target radar return signal, and    -   estimating an azimuth angle of said at least one target with        respect to said fixed reference coordinate system by means of a        tracking filter, based on said azimuth angle information and        said radar platform relative orientation, such that a        software-based motion-compensation of said radar platform is        provided.

By means of the radar system and its corresponding method presentedabove, there is no longer a need to arrange the radar antenna on anexpensive, heavy and complex mechanical motion compensating support.Consequently, a vehicle carrying a radar system according to theinvention, and thus without a mechanical motion compensation support,will show improved dynamic performance, and have higher radar functionreliability. This applies especially to radar systems arranged on marinevehicles, where the radar antenna is located at a relatively elevatedposition, where reduced weight has an increasingly positive impact onvehicle stability and roll motion, and to radar systems arranged onaeronautical vehicles, where reduced weight always has a positive impacton aeronautical performance.

According to a further advantageous aspect of the invention, saidtracking filter is configured to estimate the elevation of said at leastone target in said fixed reference coordinate system by iterativelyupdating a target elevation estimation by means of said tracking filterbased on at least two target radar return signals, each received duringseparate radar measurement scans of the same target, and each receivedat a different relative orientation of the radar platform. Knowing theorientation of the radar platform combined with at least two azimuthangle measurements of the radar antenna, each measurement taken with thefan-shaped beam in different plane at the moment of measurement, it ispossible to estimate also the elevation of a target using a 2D-antenna.The measurements in different planes are obtained by pitch and rollmotion of the radar platform, and with a time period between said atleast two measurements.

According to a further advantageous aspect of the invention, said radarsystem comprises:

-   -   a non-stable radar platform,    -   a mechanically rotated 2D-radar antenna system arranged on said        radar platform, and configured to generate a fan-shaped beam,        and to measure azimuth angle information of at least one target        radar return signal with respect to a local coordinate system of        said radar platform, and    -   radar platform orientation sensors configured to provide said        radar platform relative orientation with respect to said fixed        reference coordinate system.

According to a further advantageous aspect of the invention, saidtracking filter is configured to estimate a range and/or radial velocityof said at least one target with respect to the said radar platform.This can be done by including target parameters range and/or radialvelocity as parameters in a target state vector. Measuring andestimating range and/or radial velocity improves estimation accuracy ofthe tracking filter since more target parameter information isavailable.

According to a further advantageous aspect of the invention, the radarsystem comprises inertial sensors, like accelerometers, gyroscopes,inclinometers, or an inertial navigation system, for providing therelative orientation of said radar platform with respect to the fixedreference coordinate system. The accurate measurement of the platformorientation determines the tracking filter's possibility to accuratelycompensate for platform motion and inclination.

According to a further advantageous aspect of the invention, the radarantenna is arranged on said radar platform without mechanical motioncompensation. The radar antenna is thus strapped-down onto said platformwithout the use a servo-based motion compensating unit. Consequently,the rotation axis of the radar antenna will deviate from the orthogonalto the horizontal plane in case the platform tilts.

According to a further advantageous aspect of the invention, the radartracking filter is a nonlinear state estimation filter, for example anextended Kalman filter, or a particle filter. By estimating also theradar platform relative orientation with the tracking filter, saidfilter can concurrently take into account the uncertainty of said radarplatform relative orientation measurements, as well as the measurementsof the radar antenna system. This improves target position estimation incase of moving targets, and in case of multiple targets.

According to a further advantageous aspect of the invention, ameasurement model of said tracking filter defines a state spaceS=S_(x)·S_(θ) of a detectable target, a distribution functionp(t_(k),x_(k),θ_(k)) of the target at time t_(k) taking into account allradar return signals measured up to this time, wherein S_(θ) isdiscretized to N discrete intervals in the vertical θ-direction of afixed horizontal coordinate system, where B_(j) denotes these intervals,such that S_(θ)=U_(j)B_(j). The discretization of the elevation intervalprovides the possibility of calculating the distribution functionp(t_(k),x_(k),θ_(k)) using a normal distribution, which is piecewiseconstant for each elevation interval, even when the platform tilts andsaid distribution function no longer has a normal distribution.

According to a further advantageous aspect of the invention, thediscretization is denser where the elevation distribution P_(θ,k) ^(i)is high and less dense where the elevation distribution P_(θ,k) ^(i) issmall. This increases estimation accuracy.

According to a further advantageous aspect of the invention, the2D-radar antenna system is configured to measure target parameters(r′,ψ′,t) in said local coordinate system of said radar platform. Saidtarget parameters can be range to target (r′), azimuth angle to target(ψ′), and time (t) of target radar return signal. Said target parametersare subsequently transferred to said tracking filter, which isconfigured to produce an estimate of the state at the current time stepbased on a state estimate from a previous time step.

According to a further advantageous aspect of the invention, saidtracking filter is configured to determine coordinate transfer functionsg_(j) for all j, and transform measured target parameters (r′,ψ′,t) insaid local coordinate system to target parameters (r,ψ,Θ_(j),t) in saidfixed reference coordinate system for all different Θ_(j) by means ofsaid coordinate transfer functions g_(j).

According to a further advantageous aspect of the invention, said radartracking filter further is configured to: determine a likelihoodfunction L of the measurement at time t given the present state, andcalculate updated state estimate of the tracking filter based upon thepredicted state estimate, and the radar measurement information.

According to a further advantageous aspect of the invention, said radarsystem is located on a marine or aeronautical vehicle.

According to a further advantageous aspect of the invention, therelative orientation of said radar platform with respect to the fixedreference coordinate system is defined by roll, pitch and yaw angles ofthe radar platform.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will now be described in detail with reference tothe figures, wherein:

FIG. 1 shows a radar scanning sphere and two radar measurements at aninclined radar platform with respect to a fixed reference coordinatesystem X, Y, Z;

FIG. 2 shows the corresponding radar scanning sphere and radarmeasurements with respect to a local coordinate system X′, Y′, Z′ of theradar antenna and its platform;

FIG. 3 shows a two-dimensional side view of fan-shaped beam;

FIG. 4 shows a flowchart describing the basic steps of the stateestimation filter according to an embodiment of the invention; and

FIG. 5 shows the relation between the varying local coordinate system ofthe radar platform and the fixed horizontal coordinate system.

DETAILED DESCRIPTION

In the following only one embodiment of the invention is shown anddescribed, simply by way of illustration of one mode of carrying out theinvention.

The invention will in the following be explained when applied in amechanically rotating 2D radar system without servo based motioncompensation, and arranged on a radar platform, in particular a marinevessel. The radar uses a fan beam for both transmit and reception ofelectromagnetic energy, in particular by means of a pulse Doppler radar.The bearing, or azimuth angle, with respect to a local coordinate systemof the platform, to a detected target is measured by a sensor providingangle information of the rotating antenna with respect to the stem ofthe vessel. When the vertical axis of a vessel is orthogonal to thehorizontal plane, the target azimuth with respect to a fixed generalcoordinate system can be determined by adding the angle of the rotatingantenna at the moment of return signal with the vessel bearing fromnorth, i.e. the yaw angle.

When the vessel is tilting, the measured angle will depend not only onthe target azimuth position, but also on the target's elevation and therelative orientation of the vessel with respect to the horizontal plane.The relative orientation of the vessel in terms of roll, pitch and yawangle can be measured by means of inertial sensors, for example gyros.The target's elevation and bearing are however not known.

FIG. 1 illustrates the result when tilting the radar platform includingthe radar antenna with respect to a fixed reference coordinate system,preferably a fixed horizontal coordinate system having three axes, whereX and Y form a fixed horizontal plane and Z is orthogonal to thehorizontal plane. The radar antenna is here located at the origin 2 ofan illustrated radar scanning sphere 1 of a radar platform, which isexposed to roll and pitch motion, i.e. platform motion around the X andY axis of the horizontal coordinate system. A local coordinate systemfixed to the radar platform will thus diverge from the horizontalcoordinate system in case of roll and pitch motion. Platform motionaround the Z-axis, also called yaw motion, will not cause any errors inthe radar tracking system because this type of motion does not divergethe radar antenna's rotation axis from the orthogonal of the horizontalplane.

In FIG. 1, the solid circle 6 represents the fixed horizontal plane, thedashed circle 7 represents the platform orientation of the radarplatform at the moment of a first measurement, and the chain-dottedcircle 8 represents the platform orientation of the radar platform atthe moment of a second measurement. The platform will typically movecontinuously, and as can be seen in FIG. 1, the platform orientation atthe moment of said first and second measurements is diverged from thehorizontal coordinate system. A fixed target represented by a point 13on the radar scanning sphere 1 is detected during said first and secondmeasurement scans, and two radar fan beams 3, 4 are illustrated at thepoint of time of target detection. Said radar fan beams 3,4 are in theform of first 3 and second 4 circle sectors with their origins 2 at theorigin 2 of the radar scanning sphere 1, wherein the first circle sector3 has a first radius 9, 10 and the second circle sector 4 has a secondradius 11, 12.

FIG. 2 illustrates the same situation as FIG. 1 but with themeasurements fixed according to the local coordinate system X′, Y′, Z′of the platform instead. The solid circle 16 represents the fixed planeof the radar platform, the dashed circle 17 represents the plane ofhorizon at the moment of the first measurement, and the chain-dottedcircle 18 represents the plane of horizon at the moment of the secondmeasurement. The problem of determining the position of a detectedtarget 13 is thus made clearly visible in FIG. 2, where the first andsecond scans detect the same target at different radar antenna angles,although the target 13 is fixed in the horizontal coordinate system.

For clarification purposes, a two-dimensional side view of the first fanbeam 3 is shown in FIG. 3 at the angle of target detection in a localplatform fixed coordinate system X′, Y′ and Z′. The Z′ axis isconsequently aligned with the rotation axis of the radar antenna. Thefirst fan beam 3 is relatively tall in the elevation plane Z′ in orderto fully cover the air space, also during pitch and roll motion of theantenna.

From the above reasoning, two inventive concepts are derived:

-   -   A software-based motion-compensation of a fan-shaped beam        2D-radar antenna can replace a servo based motion-compensation        of said antenna, when a target tracking filter is provided with        information of the relative orientation of said radar platform        with respect to the horizontal coordinate system.    -   Said radar system can also determine the elevation of a target        by conducting a series of measurements of a target, when said        measurements are conducted at different relative positions of        the radar platform.

Considering that the target illustrated in FIGS. 1-3 is fixed and thatthe measurements are ideal, and that the target in a realistic scenariois moving and the measurements are inaccurate, it is advantageous toprovide a tracking filter to deal with these uncertainties. Arequirement on such a tracking filter is that it can handle nonlinearmeasurements. In the following, a non-limiting embodiment of such atracking filter is disclosed, which can estimate a target's state takinginto account target information from the radar antenna system and motioninformation of the radar platform.

Measurement and Coordinate System

Let (x,y,z)^(T) define a north-east-down (NED) Cartesian coordinatesystem. Introduce a spherical coordinate system relating to theCartesian coordinate system according to:

$\begin{matrix}\left\{ \begin{matrix}{r = \sqrt{x^{2} + y^{2} + z^{2}}} \\{\psi = {\arctan_{2}\left( {y,x} \right)}} \\{\theta = {\arctan_{2}\left( {{- z},\sqrt{x^{2} + y^{2}}} \right)}}\end{matrix} \right. & {{Equation}\mspace{14mu} (1)}\end{matrix}$

The function arctan₂ is an extension of the inverse tangent, which alsotakes into account the quadrant of (x,y) and returns an angle in theinterval (−π,π). Let (x′,y′,z′)^(T) define a Cartesian coordinate systemfixed to the radar platform, i.e. the marine vessel, where the z′-axispoints down through the vessel, the x′-axis points towards the stem, andthe y′-axis points towards starboard. A spherical coordinate system canbe introduced onto this system similar to equation 1. The sphericalcoordinate system is defined like (r′,ψ′,θ′). Let g define thetransformation between the two coordinate systems such that:

(r′, ψ′, θ′)^(T) =g(r, ψ, θ)  Equation (2)

The radar antenna and its signal processing equipment provide targetdistance information r′ and antenna angle information ψ′ measured fromthe stem of the vessel in the prime coordinate system. The beam is a fanbeam which means that the measurement can be defined according to(r′,ψ′,ξ′), where ξ′ defines the elevation area covered by the fan beam,for example ξ′ε(−π/2,π/2).

Let Z′ represent the measurements. The following is a model of themeasurements of this radar:

$\begin{matrix}{Z^{\prime} = {N\left( {\begin{pmatrix}\mu_{r} \\\mu_{\psi}\end{pmatrix},\begin{pmatrix}\sigma_{r}^{2} & 0 \\0 & {\sigma_{\psi}^{2}(\theta)}\end{pmatrix}} \right)}} & {{Equation}\mspace{14mu} (3)}\end{matrix}$

It is thus assumed that the measurements in angle and distance areindependent. The width of the beam in the ψ-direction varies also withthe elevation, which results in that the variance of ψ is a function ofθ and thus represented by σ_(ψ) ²(θ). In this filter, the measurement istransformed to the horizontal coordinate system according to:

(Z, ξ)=g ⁻¹(Z′, ξ′)  Equation (4)

Target Model and Assumptions

When detecting a target, it can be described by a state vector(X(t),Θ(t)) at the time t. Let Z_(k) represent the stochastic variablefor the observations of (X(t_(k)),Θ(t_(k))) at the time t_(k). Theresult of these observations is represented by z_(k). Let S=S_(x)·S_(θ)define the state space. Let Z(t_(k))=(Z₁, . . . Z_(k)) and z_(k)=(z₁, .. . , z_(k)). These are observations of the target made up until thetime t_(k). The distribution function of the target at time t_(k) takinginto account all the observations made up to this time is defined by:

p(t _(K) , x _(K), θ_(K))=p(X(t _(K))=x _(K), Θ(t _(K))=θ_(K) ,|Z(t_(k))=z _(K))  Equation (5)

After making the following two assumptions:

-   -   {(X(t),Θ(t)); t≧0} has Markov property;    -   Z(t_(i)) and Z(t_(j)) are independent when i≠j given        ((X(t₁)=x₁,Θ(t₁)=θ₁), . . . , (X(t_(k))=x_(k),Θ(t_(k))=θ_(k)));        equation (5) can be calculated recursively.

Initial distribution:

p(t ₀ , x ₀, θ₀)=q ₀(x ₀, θ₀), (x ₀, θ₀)εS _(x) ×S _(θ)  Equation (6)

Target Model:

The transfer function q is represented by

q _(k)(x _(k), θ_(k) |x _(k−1), θ_(k−1))=p(X(t _(k))=x _(k), Θ(t_(k))=θ_(k) |X(t _(k−1))=x _(k−1), Θ(t _(k−1))=θ_(k−1))  Equation (7)

Prediction (a priori):

A priori distribution is calculated by:

p ⁻(t _(k) , x _(k), θ_(k))=∫_(S) _(x) ∫_(S) _(θ) q _(k)(x _(k), θ_(k)|x _(k−1), θ_(k−1))p(t _(k−1) x _(k−1), θ_(k−1))dθ _(k−1) dx_(k−1)  Equation (8)

Measurement:

The likelihood function for measurement at time t_(k) is represented by

L _(k)(z _(k)|θ_(k), θ_(k))=p(Z(t _(k))=z_(k) |X(t _(k))=x _(k), Θ(t_(k))=θ_(k)), (x _(k), θ_(k))εS  Equation (9)

Filtering (a posteriori):

By means of the two assumptions made, the equation (5) can be calculatedrecursively according to:

$\begin{matrix}{{p\left( {t_{k},x_{k},\theta_{k}} \right)} = {\frac{1}{c_{k}}{L_{k}\left( {\left. z_{k} \middle| x_{k} \right.,\theta_{k}} \right)}{p^{-}\left( {t_{k},x_{k},\theta_{k}} \right)}}} & {{Equation}\mspace{14mu} (10)}\end{matrix}$

where c_(k) is a normalization constant, such that p(t_(k),.) becomes adistribution function:

c _(k) =p(z _(k))=∫_(S) _(x) ∫_(S) _(θ) L _(k)(z _(k) |x _(k), θ_(k))p⁻(t _(k), x_(k), θ_(k))dθ _(k) dx _(k)  Equation (11)

Until now, it was assumed only one target. When estimating the states ofseveral targets, it simplifies to assume that the other targets do notinterfere in the observation of a first target, so called conditionalindependency, and to assume that the target's trajectories areindependent of each other. These two assumptions make it possible todivide association and updating when several targets are tracked.

Tracking of an Air Target This tracking filter will function duringmotion of the radar platform, as well as without platform motion. If theplatform had been non-moving, a 2D-Kalman filter could have been used toestimate the state of the targets. With a moving platform however, thetarget bearing measurement depends on target elevation and platformorientation. The platform orientation is known, but target elevation isunknown and is not included in the state vector. Target elevation Θ(t)is thus added to the state vector, which now can be written (X(t),Θ(t))^(T). Let the state vector be defined by a spherical coordinatessystem having its origin on the vessel according to:

$\begin{matrix}{\begin{pmatrix}{X\left( t_{k} \right)} \\{\Theta \left( t_{k} \right)}\end{pmatrix} = \begin{pmatrix}{R\left( t_{k} \right)} \\{\overset{.}{R}\left( t_{k} \right)} \\{\Psi \left( t_{k} \right)} \\{\overset{.}{\Psi}\left( t_{k} \right)} \\{\Theta \left( t_{k} \right)}\end{pmatrix}} & {{Equation}\mspace{14mu} (12)}\end{matrix}$

Here, ψ denotes azimuth angle instead of φ since φ denotes thetransformation matrix, see equation (14). In case the state vector isdefined by a cylindrical coordinates system instead, the following statevector is provided:

$\begin{matrix}{\begin{pmatrix}{X\left( t_{k} \right)} \\{\Theta \left( t_{k} \right)}\end{pmatrix} = \begin{pmatrix}{R\left( t_{k} \right)} \\{\overset{.}{R}\left( t_{k} \right)} \\{Y\left( t_{k} \right)} \\{\overset{.}{Y}\left( t_{k} \right)} \\{\Theta \left( t_{k} \right)}\end{pmatrix}} & {{Equation}\mspace{14mu} (13)}\end{matrix}$

A cylindrical coordinate system should be oriented such that thecylinder axis is orthogonal to the direction of the target. To track anair target, a motion model of the target is needed. In case the targetis limited to a land- or see based object, or if the radar platform wasfixed with respect to the horizontal plane, a two dimensional Kalmanfilter could have been adopted. But when tracking an air target, alsothe elevation Θ of the target must be estimated and a three dimensionaltarget motion model will be derived. It is assumed that the targets movein straight trajectories. Hence, the motion model of the target is:

$\begin{matrix}{\begin{pmatrix}{X\left( t_{k} \right)} \\{\Theta \left( t_{k} \right)}\end{pmatrix} = {{\begin{pmatrix}\Phi_{k} & 0 \\0 & 1\end{pmatrix}\begin{pmatrix}{X\left( t_{k - 1} \right)} \\{\Theta \left( t_{k - 1} \right)}\end{pmatrix}} + \begin{pmatrix}b_{x,k} \\b_{\theta,k}\end{pmatrix} + \begin{pmatrix}w_{x,k} \\w_{\theta,k}\end{pmatrix}}} & {{Equation}\mspace{14mu} (14)}\end{matrix}$

where b_(k) is a term reflecting the vessel's own displacement betweent_(k−1) and t_(k). For spherical coordinates φ_(k) denotes the Jacobianfor a transfer function, which describes a straight trajectory, andb_(k) comprises then also constant part of the linearization. Theprocess noise is assumed to follow the normal distribution withexpectation value zero, i.e. w_(x,k)˜N(0,Q_(k)), and w_(x,θ) has adistribution function denoted h, which is further described later in thetext.

The distribution function p(t_(k),x_(k),θ_(k)):

In case the vessel does not experience any relative motion with respectto the horizontal coordinate system, p(t_(k),x_(k)) would have normaldistribution. However, in case the vessel does experience relativemotion, said distribution no longer applies. To calculatep(t_(k),x_(k),θ_(k)), it would be possible to discretize S_(x) andS_(θ). This approach would however need too much computational effort toachieve required accuracy. Instead, it is possible to limit thediscretization to N discrete intervals in the θ-direction only, whereB_(j) denotes these intervals, and where |B_(j)| is the length of saidinterval B_(j). The discretization is selected such thatS_(θ)=U_(j)B_(j). The distribution function θ_(k)→p(t_(k),x_(k),θ_(k))is thus assumed to be piecewise constant for each interval. In theremaining coordinates, the distribution functionx_(k)→p(t_(k),x_(k),θ_(k)) is assumed to have normal distribution. Thedistribution function is thus defined according to:

$\begin{matrix}{{p\left( {t_{k},x_{k},\theta_{k}} \right)} = {\sum\limits_{j = 1}^{N}{{{\eta \left( {x_{k},\mu_{x,k}^{j},\Sigma_{x,k}^{j}} \right)} \cdot \frac{1}{B_{j}}}{P_{\theta,k}^{j} \cdot \chi}\; {B_{j}\left( \theta_{k} \right)}}}} & {{Equation}\mspace{14mu} (15)}\end{matrix}$

Here, η_(i)(x,μΣ) the normal distribution with expectation value μ andvariance Σ. P_(θ) ^(j) denotes the likelihood that the target is withinthe interval B_(j), and μ_(e) ^(j) defines the centre of the intervalB_(j). The marginal distributions are defined by the following twoequations:

$\begin{matrix}\begin{matrix}{{p\left( {t_{k},x_{k}} \right)} = {\int_{S_{\theta}}^{\;}{{p\left( {t_{k},x_{k},\theta_{k}} \right)}{\theta_{k}}}}} \\{= {\sum\limits_{j = 1}^{N}{{\eta \left( {x_{k},\mu_{x,k}^{j},\Sigma_{x,k}^{j}} \right)} \cdot P_{\theta,k}^{j}}}}\end{matrix} & {{Equation}\mspace{14mu} (16)} \\\begin{matrix}{{p\left( {t_{k},\theta_{k}} \right)} = {\int_{S_{x}}^{\;}{{p\left( {t_{k},x_{k},\theta_{k}} \right)}{x_{k}}}}} \\{= {\sum\limits_{j = 1}^{N}{\frac{P_{\theta,k}^{j}}{B_{j}}\chi \; {B_{j}(\theta)}}}}\end{matrix} & {{Equation}\mspace{14mu} (17)}\end{matrix}$

The target location is measured by the radar in spherical coordinates(range, azimuth). Tracking in spherical coordinates is however difficultsince motion of constant velocity targets (straight lines) will causeacceleration terms in all coordinates. A simple solution to this problemis to track in horizontal coordinates. Hence, the measurement of thetarget position is transformed to the horizontal coordinate system.Since only distance and detection angle are measured, the measurementwill cross several different intervals B_(j). For each measuredinterval, ψ^(j) must be determined. This is performed by adding a thirdcoordinate to the measurement μ′_(θ) ^(j), and by selecting this suchthat the transformed measurement lies on the elevation μ_(θ) ^(j). Sinceg is a bijection, there is single μ′_(θ) ^(j) that fulfils this. Hence,according to equation (4):

(z ^(j), μ_(θ) ^(j))=g ⁻¹(z′ ^(j), μ′_(θ) ^(j))  Equation (18)

Equation (18) is used to determine the likelihood function for themeasurement. Now, all necessary assumptions are ready, and in thefollowing, prediction and filtering will be derived. Prediction isperformed according to equation (14) by:

q _(k)(x _(k), θ_(k) |x _(k−1), θ_(k−1))=Σ_(j=1) ^(N)η(x _(k), φ_(k) x_(k−1) +b _(x,k) ^(j) , Q _(x,k) ^(j))·h(θ_(k), θ_(k−1) +b _(θ,k) ^(j) ,Q _(θ,k) ^(j))χB _(j)(θ_(k−1))  Equation (19)

Here, b_(x,k) ^(j) denotes a distance traveled by the own vessel plus anadditional linearization contribution in case the target is tracked bymeans of spherical coordinates. In an analogue manner, b_(θ,k) ^(j)denotes a term for the distance moved of the origin of the coordinatesystem due to the motion of the own vessel, and h denotes a function ofthe target in the elevation direction. θ_(k−1)+b_(θ,k) ^(j) denotes theexpectation value, and Q_(θ,k) ^(j) denotes some type of diffusion termdepending on choice of function. For example, the following functiondefines target elevation motion in case said target moves according to auniform distribution:

$\begin{matrix}{{h\left( {\theta_{k},{\theta_{k - 1} + b_{\theta,k}^{j}},Q_{\theta,k}^{j}} \right)} = {\frac{1}{D_{j}}\chi \; {D_{j}\left( {\theta_{k} - \left( {\theta_{k - 1} + b_{\theta,k}^{j}} \right)} \right)}}} & {{Equation}\mspace{14mu} (20)}\end{matrix}$

The length of the interval |D_(j)| depends on the target maximum speedin elevation direction.

A state estimation prediction can now be made in two steps. For alldirections except θ-direction, q describes the update for a normalKalman filter:

p ⁻(t _(k) ,x _(k), θ_(k))=∫_(S) _(x) ∫_(S) _(θ) q _(k)(x _(k), θ_(k) |x_(k−1), θ_(k−1))p(t _(k−1) x _(k−1), θ_(k−1))dθ _(k−1) dx _(k−1)=Σ_(j=1)^(N) ∫_(S) _(x) ∫_(S) _(θ) η(x _(k), φ_(k) x _(k−1) +b _(x,k) ^(k) , Q_(x,k) ^(j))·h(θ_(k), θ_(k−1) +b _(θ,k) ^(j) , Q _(θ,k) ^(j))·η(x_(k−1), μ_(x,k−1) ^(j), Σ_(x,k−1) ^(j))·1/|b _(j) |P _(θ,k−1) ^(j) ·χB_(j)(θ_(k−1))dθ _(k−1) dx _(k−1)=Σ_(j=1) ^(N) ∫_(S) _(θ) η(x _(k),{tilde over (μ)}_(x,k) ^(−,j), {tilde over (Σ)}_(x,k) ^(−,j))1/|B _(j)|P _(θ,k−1) ^(j) χB _(j)(θ_(k−1))·h(θ_(k), θ_(k−1) +b _(θ,k) ^(j) , Q_(θ,k) ^(j))dθ _(k−1)  Equation (21)

The variables marked with tilde are those received by the Kalman filterfor X|(ΘεB_(j)), i.e.:

{tilde over (μ)}_(x,k) ^(−,j)=φ_(k)μ_(x,k−1) +b _(x,k) ^(j)  Equation(22)

{tilde over (Σ)}_(x,k) ^(−,j)=φ_(k)Σ_(x,k−1) ^(j)φ_(k) ^(T) +Q _(x,k)^(j)  Equation (23)

Assuming h according (20), then (21) will have the form:

$\begin{matrix}{{p^{-}\left( {t_{k},x_{k},\theta_{k}} \right)} = {\sum\limits_{j = 1}^{N}{{\eta \left( {x_{k},{\overset{\sim}{\mu}}_{x,k}^{- {,j}},{\overset{\sim}{\Sigma}}_{x,k}^{- {,j}}} \right)}P_{\theta,{k - 1}}^{j}\frac{1}{{B_{j}}{D_{j}}}\chi \; B_{j}*\chi \; {D_{j}\left( {\theta_{k} - b_{\theta,k}^{j}} \right)}}}} & {{Equation}\mspace{14mu} (24)}\end{matrix}$

This function must be approximated with a function piecewise constant inthe θ-direction, and having a normal distribution in remainingdirections, i.e. a function like (15). By introducing the term a_(ij)according to:

$\begin{matrix}{{\int_{B_{i}}^{\;}{{p^{-}\left( {t_{k},x_{k},\theta_{k}} \right)}{\theta_{k}}}} = {\sum\limits_{j = 1}^{N}{{\eta \left( {x_{k},{\overset{\sim}{\mu}}_{x,k}^{- {,j}},{\overset{\sim}{\Sigma}}_{x,k}^{- {,j}}} \right)}{P_{\theta,{k - 1}}^{j} \cdot a_{ij}}}}} & {{Equation}\mspace{14mu} (25)}\end{matrix}$

Said term a_(ij) represents the likelihood for transfer betweendifferent intervals, i.e. the probability that a target within intervalB_(j) should have moved to interval B_(i) since the last measurement att_(k−1). The distribution is received by:

$\begin{matrix}\begin{matrix}{P_{\theta,k}^{- {,i}} = {\int_{S_{x}}^{\;}{\int_{B_{i}}^{\;}{{p^{-}\left( {t_{k},x_{k},\theta_{k}} \right)}{\theta_{k}}{x_{k}}}}}} \\{= {\sum\limits_{j = 1}^{N}{P_{\theta,{k - 1}}^{j} \cdot a_{ij}}}}\end{matrix} & {{Equation}\mspace{14mu} (26)}\end{matrix}$

The expectation values are received by:

$\begin{matrix}\begin{matrix}{{P_{\theta,k}^{- {,i}} \cdot \mu_{x,k}^{- {,i}}} = {\int_{S_{x}}^{\;}{\int_{B_{i}}^{\;}{x_{k}{p^{-}\left( {t_{k},x_{k},\theta_{k}} \right)}{\theta_{k}}{x_{k}}}}}} \\{= {\sum\limits_{j = 1}^{N}{{\overset{\sim}{\mu}}_{x,k}^{- {,j}}{P_{\theta,{k - 1}}^{j} \cdot a_{ij}}}}}\end{matrix} & {{Equation}\mspace{14mu} (27)}\end{matrix}$

Finally, the covariance matrixes are calculated by:

$\begin{matrix}\begin{matrix}{{P_{\theta,k}^{- {,i}} \cdot \Sigma_{x,k}^{- {,i}}} = {\int_{S_{x}}^{\;}{\int_{B_{i}}^{\;}{\left( {x_{k} - \mu_{x,k}^{- {,i}}} \right)\left( {x_{k} - \mu_{x,k}^{- {,i}}} \right)^{T}}}}} \\{{{p^{-}\left( {t_{k},{x_{k}\theta_{k}}} \right)}{\theta_{k}}{x_{k}}}} \\{= {\int_{S_{x}}^{\;}{\int_{B_{i}}^{\;}\left( {x_{k} - {\overset{\sim}{\mu}}_{x,k}^{- {,j}} + \left( {{\overset{\sim}{\mu}}_{x,k}^{- {,j}} - \mu_{x,k}^{- {,i}}} \right)} \right)}}} \\{{\left( {x_{k} - {\overset{\sim}{\mu}}_{x,k}^{- {,j}} + \left( {{\overset{\sim}{\mu}}_{x,k}^{- {,j}} - \mu_{x,k}^{- {,i}}} \right)} \right)^{T}{p^{-}\left( {t_{k},x_{k},\theta_{k}} \right)}}} \\{{{\theta_{k}}{x_{k}}}} \\{= {\sum\limits_{j = 1}^{N}{a_{ij}{P_{\theta,{k - 1}}^{j} \cdot \left\lbrack {{\overset{\sim}{\Sigma}}_{x,k}^{- {,j}} + \left( {{\overset{\sim}{\mu}}_{x,k}^{- {,j}} - \mu_{x,k}^{- {,i}}} \right)} \right.}}}} \\\left. \left( {{\overset{\sim}{\mu}}_{x,k}^{- {,j}} - \mu_{x,k}^{- {,i}}} \right)^{T} \right\rbrack\end{matrix} & {{Equation}\mspace{14mu} (28)}\end{matrix}$

At this point, the a priori distribution has been determined and ameasurement based state estimate update can be performed: State estimateupdates shall be performed using (10). Let z_(k) ^(j) denote themeasurement at time point t_(k) transferred to the coordinate systemused for state estimation of the target. The calculation of z_(k) ^(j)is determined by (18). Let R_(k) ^(j) be the covariance matrix for thetransferred measurement. Due to the orientation of the vessel, and thelimited elevation coverage of the radar antenna, a measurement can notalways be transferred to all B_(j). Let A_(k) denote the subset of {1, .. . , N} where measurements are available:

A _(k) ={jε{1, . . . , N}; ∃z _(k) ^(j), θ′_(k) ^(j) lies within theantenna coverage}  Equation (29)

The likelihood function L_(k) is according the measurement model definedaccording to:

$\begin{matrix}{{L_{k}\left( {\left. z_{k} \middle| x_{k} \right.,\theta_{k}} \right)} = {\sum\limits_{j \in A_{k}}{{\eta \left( {z_{k}^{j},{M\; x_{k}},R_{k}^{j}} \right)}{\chi_{B_{j}}(\theta)}}}} & {{Equation}\mspace{14mu} (30)}\end{matrix}$

The update will then have the form:

$\begin{matrix}{{p\left( {t_{k},x_{k},\theta_{k}} \right)} = {\frac{1}{c}{L_{k}\left( {\left. z_{k} \middle| x_{k} \right.,\theta_{k}} \right)}{p^{-}\left( {t_{k},x_{k},\theta_{k}} \right)}}} & {{Equation}\mspace{14mu} (31)}\end{matrix}$

Since the distribution function is piecewise constant in θ, this can bewritten as:

$\begin{matrix}\begin{matrix}{{p\left( {t_{k},x_{k},\theta_{k}} \right)} = {\sum\limits_{j \in A_{k}}{{{\eta \left( {x_{k},\mu_{x,k}^{j},\sum_{x,k}^{j}} \right)} \cdot \frac{1}{B_{j}}}{P_{\theta,k}^{j} \cdot {\chi_{B_{j}}(\theta)}}}}} \\{= {\frac{1}{c}{\sum\limits_{j \in A_{k}}{{\eta \left( {z_{k}^{j},{M\; x_{k}},R_{k}^{j}} \right)}{{\eta \left( {x_{k},\mu_{x,k}^{- {,j}},\Sigma_{x,k}^{- {,j}}} \right)} \cdot}}}}} \\{{\frac{1}{B_{j}}{P_{\theta,k}^{- {,j}} \cdot {\chi_{B_{j}}(\theta)}}}}\end{matrix} & {{Equation}\mspace{14mu} (32)}\end{matrix}$

The calculation of (32) is made in two steps. Firstly, each intervalB_(j) can be calculated separately by (32). The normalization c isdetermined starting from:

η(x _(k), μ_(x,k) ^(j), Σ_(x,k) ^(j))c _(j)=η(z _(k) , Mx _(k) , R _(k)^(j))η(x _(k), μ_(x,k) ^(−,j), Σ_(x,k) ^(−,j))  Equation (33)

This is a normal update of a Kalman filter. To determine c_(j), which isneeded to determine the new distribution in θ-direction, one starts outwith the following identity (Bayes rule):

p(x _(k) |z _(k) ^(j))p(z _(k) ^(j))=p(z _(k) ^(j) | _(k))p(x_(k))  Equation (34))

Both p(x_(k)|z_(k) ^(j)) and p(z_(k) ^(j)) must be determined. The termp(x_(k)|z_(k) ^(j)) is known from the Kalman filter update, and thusalready described in many sources, and can be derived by the well-known“matrix inversion lemma”. The term p(x_(k)|z_(k) ^(j))=c_(j) is neededto determine c together with

$\frac{1}{B_{j}}{P_{\theta,k}^{- {,j}}.}$

The transferred measurement can thus be described by Z_(k)^(j)=MX(t_(k))+ε_(k) ^(j), where ε_(k) ^(j)˜N(0,R_(k) ^(j)). This givesE[Z_(k) ^(j)]=Mμ_(x,k) ^(−,j) and Var[Z_(k) ^(j)]=MZ_(x,k)^(−,j)M^(T)+R_(k) ^(j). Hence:

p(z _(k) ^(j))=η(z _(k) ^(j) , Mμ _(z,k) ^(−,j) , S _(k) ^(j))  Equation(35)

where

S _(k) ^(j) =MΣ _(x,k) ^(−,j) M ^(T) +R _(k) ^(j)  Equation (36)

New expectation values and covariance matrixes can now be calculated by:

μ_(x,k) ^(j)=μ_(x,k) ^(−,j)+Σ_(x,k) ^(−,j) M ^(T)(S ^(j))⁻¹(z _(k) ^(j)−Mμ _(z,k) ^(−,j))  Equation (37)

and

Σ_(x,k) ^(j)=Σ_(x,k) ^(−,j)−Σ_(x,k) ^(−,j) M ^(T)(S ^(j))⁻¹ MΣ _(x,k)^(−,j)  Equation (38)

Finally, P_(θ,k) ^(i) will be calculated. First, the normalizationconstant c is determined by introducing (33) and (35) in equation (32)and integrating over (S_(x)×S_(θ)):

$\begin{matrix}{{\int_{S_{x}}{\int_{S_{\theta}}{{{cp}\left( {t_{k},x_{k},\theta_{k}} \right)}{\theta_{k}}{x_{k}}}}} = \left. {\int_{S_{x}}{\int_{S_{\theta}}{\Sigma_{j \in A_{k}}{\eta \left( {z_{k}^{j},{M\; \mu_{x,k}^{- {,j}}},S_{k}^{j}} \right)}{{\eta \left( {x_{k},\mu_{x,k}^{j},\Sigma_{x,k}^{j}} \right)} \cdot \frac{1}{B_{j}}}{P_{\theta,k}^{- {,j}} \cdot {\chi_{B_{j}}(\theta)}}{\theta_{k}}{x_{k}}}}}\Leftrightarrow \right.} & (39) \\{\mspace{79mu} {c = {\Sigma_{j \in A_{k}}{\eta \left( {z_{k}^{j},{M\; \mu_{x,k}^{- {,j}}},S_{k}^{j}} \right)}P_{\theta,k}^{- {,j}}}}} & (40)\end{matrix}$

Now, P_(θ,k) ^(i) can be calculated by integrating over B_(i) inequation (39) instead of S_(θ). Then, P_(θ,k) ^(i) is calculatedaccording to:

$\begin{matrix}{P_{\theta,k}^{i} = \frac{P_{\theta,k}^{- {,i}}{\eta \left( {z_{k}^{i},{M\; \mu_{x,k}^{- {,i}}},S^{i}} \right)}}{\Sigma_{j \in A_{k}}P_{\theta,k}^{- {,j}}{\eta \left( {z_{k}^{j},{M\; \mu_{x,k}^{- {,j}}},S^{j}} \right)}}} & (41)\end{matrix}$

This ends the state estimation update and the result can be presented.It is possible to calculate the expectation value of (X(t_(k))^(T),Θ(t_(k)))^(T) by first calculating μ_(θ,k):

$\begin{matrix}\begin{matrix}{\mu_{\theta,k} = {\int_{S_{x}}{\int_{S_{\theta}}{\theta_{k}{p\left( {t_{k},x_{k},\theta_{k}} \right)}{\theta_{k}}{\theta}\; x_{k}}}}} \\{= {\sum\limits_{j = 1}^{N}{\int_{S_{\theta}}{\theta_{k}{P_{\theta,k}^{j} \cdot \frac{1}{B_{j}} \cdot {\chi_{B_{j}}(\theta)}}{\theta_{k}}}}}} \\{= {\sum\limits_{j = 1}^{N}{\mu_{\theta,k}^{j}P_{\theta,k}^{j}}}}\end{matrix} & (42)\end{matrix}$

and subsequently μ_(x,k):

$\begin{matrix}\begin{matrix}{\mu_{x,k} = {\int_{S_{x}}{\int_{S_{\theta}}{x_{k}{p\left( {t_{k},x_{k},\theta} \right)}{\theta_{k}}{x_{k}}}}}} \\{= {\sum\limits_{j = 1}^{N}{\int_{S_{\theta}}{\mu_{x,k}^{j}{P_{\theta,k}^{j} \cdot \frac{1}{B_{j}} \cdot {\chi_{B_{j}}(\theta)}}{\theta_{k}}}}}} \\{= {\sum\limits_{j = 1}^{N}{\mu_{x,k}^{j}P_{\theta,k}^{j}}}}\end{matrix} & (43)\end{matrix}$

In certain situations, it might also be of interest to obtain thecovariance matrix:

$\begin{matrix}\begin{matrix}{{{Var}\left\lbrack \begin{pmatrix}{X\left( t_{k} \right)} \\{\Theta \left( t_{k} \right)}\end{pmatrix} \right\rbrack} = {\int_{S_{x}}{\int_{B_{i}}{\left( {\begin{pmatrix}x_{k} \\\theta_{k}\end{pmatrix} - \begin{pmatrix}\mu_{x,k} \\\mu_{\theta,k}\end{pmatrix}} \right){\left( {\begin{pmatrix}x_{k} \\\theta_{k}\end{pmatrix} - \begin{pmatrix}\mu_{x,k} \\\mu_{\theta,k}\end{pmatrix}} \right)^{T} \cdot}}}}} \\{{{p\left( {t_{k},x_{k},\theta_{k}} \right)}{\theta_{k}}{x_{k}}}} \\{= {\sum\limits_{j = 1}^{N}{P_{\theta,k}^{j}\begin{pmatrix}\begin{matrix}{\Sigma_{x,k}^{j} + \left( {\mu_{x,k}^{j} - \mu_{x,k}} \right)} \\\left( {\mu_{x,k}^{j} - \mu_{x,k}} \right)^{T}\end{matrix} & \begin{matrix}\left( {\mu_{x,k}^{j} - \mu_{x,k}} \right) \\\left( {\mu_{\theta,k}^{j} - \mu_{\theta,k}} \right)\end{matrix} \\\begin{matrix}\left( {\mu_{\theta,k}^{j} - \mu_{\theta,k}} \right) \\\left( {\mu_{x,k}^{j} - \mu_{x,k}} \right)^{T}\end{matrix} & \begin{matrix}{{{B_{j}}^{2}/12} +} \\\begin{pmatrix}{\mu_{\theta,k}^{j} -} \\\mu_{\theta,k}\end{pmatrix}^{2}\end{matrix}\end{pmatrix}}}}\end{matrix} & {{Equation}\mspace{14mu} (41)}\end{matrix}$

Equation (41) ends the derivation of a tracking filter, which issuitable to be implemented in a radar system. The steps and equationsneeded to transform the radar measurement to an output display unit arepresented below in relation to the flowchart of FIG. 4, whichillustrates the basic steps of the calculation according to theinventive method.

In a first step 41, the radar antenna system performs signal processingon the return signals received by the radar antenna. If a target isdetected, its target parameters are estimated based upon the returnsignal. The target parameters included in this embodiment are: distanceto target(r′), target detection angle (ψ′), and point of time of thereturn signal corresponding to said target detection. Said targetparameters (r′,ψ′,t) are determined in the local platform basedcoordinate system of the radar system, and are subsequently transferredto step 42.

In a second step 42, a state estimation prediction of the statevariables of the Kalman filter is performed at the current time stepbased on the previous estimated state of the filter. Equations (22),(23), (26), (27) and (28) determine said state estimation prediction,and they are summarized below:

{tilde over (μ)}_(x,k) ^(−,j)=φ_(k)μ_(x,k−1) +b _(x,k) ^(j)

{tilde over (Σ)}_(x,k) ^(−,j)=φ_(k)Σ_(x,k−1) ^(j)φ_(k) ^(T) +Q _(x,k)^(j)

Determine the transfer likelihood terms a_(ij) according to (25), whereP_(θ,k) ^(−,j) is the elevation distribution:

$P_{\theta,k}^{- {,i}} = {\sum\limits_{j = 1}^{N}{P_{\theta,{k - 1}}^{j} \cdot a_{ij}}}$

Expectation values μ_(x,k) ^(−,j):

${P_{\theta,k}^{- {,i}} \cdot \mu_{x,k}^{- {,i}}} = {\sum\limits_{j = 1}^{N}{{\overset{\sim}{\mu}}_{x,k}^{- {,j}}{P_{\theta,{k - 1}}^{j} \cdot a_{ij}}}}$

Covariance matrixes Σ_(x,k) ^(−,j):

${P_{\theta,k}^{- {,i}} \cdot \Sigma_{x,k}^{- {,i}}} = {\sum\limits_{j = 1}^{N}{a_{ij}{P_{\theta,{k - 1}}^{j} \cdot \left\lbrack {{\overset{\sim}{\Sigma}}_{x,k}^{- {,j}} + {\left( {{\overset{\sim}{\mu}}_{x,k}^{- {,j}} - \mu_{x,k}^{- {,j}}} \right)\left( {{\overset{\sim}{\mu}}_{x,k}^{- {,j}} - \mu_{x,k}^{- {,i}}} \right)^{T}}} \right\rbrack}}}$

In step 43, coordinate transfer functions g_(j) are determined for allj, wherein j denotes the discretiziced intervals of the radar coveragein θ-direction, i.e. the elevation direction in the horizontalcoordinate system. B_(j) denotes said intervals. Said coordinatetransfer functions g_(j) transform each measurement (r′,ψ′,t) of thelocal platform based coordinate system to (r,ψ,Θ_(j),t) of thehorizontal coordinate system for all different Θ_(j). Information ofvessel orientation (pitch, roll and yaw) at the point of time t isnecessary to derive these transformations, which orientation is obtainedfor example by an inertial navigation system of the radar platform.

In step 44, the radar observation of step 41 is transformed using thetransfer functions g_(j), and the corresponding likelihood function L ofthe measurement at time t is determined given the present state.

In step 45, a state estimation update of the state variables isperformed based upon the predicted state estimate of the target, and theradar measurement information. Equations (36), (37), (38) and (41)determine said calculations of the filtering, and they are summarizedbelow: Expectation values μ_(x,k) ^(j):

μ_(x,k) ^(j)=μ_(x,k) ^(−,j)+Σ_(x,k) ^(−,j) M ^(T)(S ^(j))⁻¹(z _(k) ^(j)−Mμ _(x,k) ^(−,j))

and covariance matrixes Σ_(x,k) ^(j):

Σ_(x,k) ^(j)=Σ_(x,k) ^(−,j)−Σ_(x,k) ^(−,j) M ^(T)(S ^(j))−1 MΣ _(x,k)^(−,j)

where

S _(k) ^(j) =MΣ _(x,k) ^(−,j) M ^(T) +R _(k) ^(j)

Elevation distribution, P_(θ,k) ^(i):

$P_{\theta,k}^{i} = \frac{P_{\theta,k}^{- {,i}}{\eta \left( {z_{k}^{i},{M\; \mu_{x,k}^{- {,i}}},S^{i}} \right)}}{\sum\limits_{j = 1}^{N}{P_{\theta,k}^{- {,j}}{\eta \left( {z_{k}^{j},{M\; \mu_{x,k}^{- {,j}}},S^{j}} \right)}}}$

In step 46, the result of the filtering can be derived by calculating:expectation value:

${E\left\lbrack \begin{pmatrix}{X\left( t_{k} \right)} \\{\Theta \left( t_{k} \right)}\end{pmatrix} \right\rbrack} = {\sum\limits_{j = 1}^{N}{P_{\theta,k}^{j}\begin{pmatrix}\mu_{x,k}^{j} \\\mu_{\theta,k}^{j}\end{pmatrix}}}$

And variance:

${{Var}\left\lbrack \begin{pmatrix}{X\left( t_{k} \right)} \\{\Theta \left( t_{k} \right)}\end{pmatrix} \right\rbrack} = {\sum\limits_{j = 1}^{N}{P_{\theta,k}^{j}\begin{pmatrix}{\sum_{x,k}^{j}{{+ \left( {{\mu_{x,k}^{j} - \mu_{x,k}}} \right)}\left( {\mu_{x,k}^{j} - \mu_{x,k}} \right)^{T}}} & {\left( {\mu_{x,k}^{j} - \mu_{x,k}} \right)\left( {\mu_{\theta,k}^{j} - \mu_{\theta,k}} \right)} \\{\left( {\mu_{\theta,k}^{j} - \mu_{\theta,k}} \right)\left( {\mu_{x,k}^{j} - \mu_{x,k}} \right)^{T}} & {{{B_{j}}^{2}/12} + \left( {\mu_{\theta,k}^{j} - \mu_{\theta,k}} \right)^{2}}\end{pmatrix}}}$

Finally, in step 47, the calculated result can be presented by anysuitable means, for example on a display.

In FIG. 5, the varying local coordinate system of the radar platform ψ′,θ′, as well as the static horizontal coordinate system ψθ isillustrated, and how they correlate. The radar system makes observationsof a target in local platform based coordinate system. The parameters ofa detected target for a 2D fan beam radar are the target distance r′,and the target bearing ψ′. No information is however available about theelevation θ′. Hence, target bearing ψ′ in the local coordinate system isextended 51 in the elevation direction to indicate all possibleelevation locations of said target within the elevation scope of theradar beam, all having the identical target bearing ψ′ in the localcoordinate system. Note here that elevation direction of the radarplatform orientation θ′ differs from the elevation direction of thehorizontal coordinate system θ, because of the movement of the vessel onwhich the radar antenna is located. Said elevation scope of the radarbeam is subsequently divided into N discrete intervals B_(j), j=1 . . .N by the tracking filter in elevation direction of the horizontalcoordinate system θ, such that the union of B_(j) covers the entireelevation scope. The centre of each said interval B_(j) is denotedθ_(j). Given that a target is located at elevation θ_(j), the bearing ψof the target in the horizontal coordinate system can be estimated. Aset of estimations ψ|θ_(j) for j=1 . . . N are thus provided. The circle52 indicates the relationship between ψ′ and ψ|θ_(j).

When the platform does not move, the measured angle will be the same inall elevation bands, B_(j), which updates the tracking filter. Thetracking filter will thus work as a 2D Kalman filter in such asituation.

To increase accuracy, uniform discretization should be avoided. Thepoints of discretizations should be selected to such that that they liemore dense where P_(θ,k) ^(i) is high and less dense where P_(θ,k) ^(i)is small. This can be achieved after each filtering loop for example bydividing those intervals B_(i) in two parts, which corresponds toP_(θ,k) ^(i)>threshold.

The calculations described above will only be performed for thoseintervals B_(j) where P_(θ,k) ^(j)≠0.

Tracking of a Sea- or Land Based Target

Since sea and land based targets do not move in the elevation direction,the target elevation is always known. Due to this, equation (4) can beused to transform the measurement to a horizontal coordinate system,wherein a known ξ′ implies that ξ=0. Hence, the measurement Z′ can betransferred to horizontal coordinate system according to:

(Z, 0)=g ⁻¹(Z′,ξ′)  Equation (44)

The state of the target can now be estimated by means of a 2D Kalmanfilter.

The disclosed radar system can simultaneously track multiple targets,and the invention is capable of modification in various obviousrespects, all without departing from the scope of the appended claims.Accordingly, the drawings and the description thereto are to be regardedas illustrative in nature, and not restrictive.

The term relative orientation of the radar platform is throughout thisdisclosure considered to represent the relative orientation of the radarplatform's local coordinate system with respect to said horizontalcoordinate system. The relative orientation is defined in terms of roll,pitch and yaw angles.

Pitch, roll and yaw angles measure the absolute attitude angles of avessel relative to the horizon/true north. These are defined as:

Pitch angle: Angle of x′-axis of the vessel relative to horizon;Roll angle: Angle of y′-axis of the vessel relative to horizon;Yaw angle: Angle of x′-axis of the vessel relative to North;where the x′-axis points towards the stem of the vessel, and the y′-axispoints towards starboard of the vessel.

The term radar platform is considered to signify a vehicle body, forexample a marine vessel or an aircraft, which rotatably supports a radarantenna. In case the radar antenna is stabilized by a servo based motioncompensating support as in the prior art, the rotating axis of the radarantenna will constantly be substantially orthogonal to the horizontalplane despite platform roll and pitch motion. In case of a puresoftware-based motion-compensation of the antenna according to theinvention however, the rotating axis of the radar antenna willconstantly be substantially parallel to the z-axis of the vehicle, i.e.the radar antenna will have a varying relative orientation with respectto the horizontal coordinate system in case of platform roll and pitchmotion.

The term “narrow-fan type radar” is considered to represent a radarsystem having an antenna, which produces a main beam having a narrowbeam width in the horizontal plane, often around 1°, and a wider beamwidth in the vertical plane, in particular 20°-100°.

1. A radar system for detecting and tracking at least one targetutilizing a mechanically rotated two-dimensional radar antenna systemwith a fan-shaped beam, arrangeable on a non-stable radar platform, theradar system comprising: a tracking filter configured to estimate anazimuth angle of said at least one target with respect to a fixedreference coordinate system, based on: azimuth angle information of atleast one target radar return signal measured by means of said radarantenna system with respect to a local coordinate system of said radarplatform, radar platform relative orientation with respect to said fixedreference coordinate system at the time of said at least one targetradar return signal, such that a software-based motion-compensation ofsaid radar platform is provided, wherein said tracking filter isconfigured to estimate the elevation of said at least one target in saidfixed reference coordinate system by iteratively updating a targetelevation estimation by means of said tracking filter based on at leasttwo target radar return signals, each received during separate radarmeasurement scans of the same target, and each received at a differentrelative orientation of the radar platform.
 2. The radar systemaccording to claim 1, further comprising: a radar platform, amechanically rotated 2D-radar antenna system arranged on said radarplatform, and configured to generate a fan-shaped beam, and to measureazimuth angle information of at least one target radar return signalwith respect to a local coordinate system of said radar platform, andradar platform orientation sensors configured to provide said radarplatform relative orientation with respect to said fixed referencecoordinate system.
 3. The radar system according to claim 1, whereinsaid radar platform orientation sensors comprise at least one ofinertial sensors, accelerometers, gyroscopes, inclinometers, or aninertial navigation system, for providing the relative orientation ofsaid radar platform with respect to said fixed reference coordinatesystem.
 4. The radar system according to claim 1, wherein the radartracking filter is a nonlinear state estimation filter or a particlefilter.
 5. A method for detecting and tracking at least one targetutilizing a mechanically rotated two-dimensional radar antenna systemwith a fan-shaped beam, arrangeable on a platform, the methodcomprising: obtaining azimuth angle information of at least one targetradar return signal measured by said radar antenna system with respectto a local coordinate system of said radar platform, obtaining radarplatform relative orientation with respect to a fixed referencecoordinate system at the time of said at least one target radar returnsignal, estimating an azimuth angle of said at least one target withrespect to said fixed reference coordinate system utilizing a trackingfilter, based on said azimuth angle information and said radar platformrelative orientation, such that a software-based motion-compensation ofsaid radar platform is provided, and estimating the elevation of said atleast one target in said fixed reference coordinate system utilizingsaid tracking filter, by iteratively updating a target elevationestimation utilizing said tracking filter based on at least two targetradar return signals, each received during separate radar measurementscans of the same target, and each received at a different relativeorientation of the radar platform.
 6. The method according to claim 5,wherein the relative orientation of said radar platform with respect tothe fixed reference coordinate system is defined by roll, pitch and yawangles of the radar platform.
 7. The method according to claim 5,wherein the relative orientation of said radar platform with respect tothe fixed reference coordinate system is obtained utilizing inertialsensors, accelerometers, gyroscopes, inclinometers, or an inertialnavigation system.
 8. The method according to claim 5, wherein themeasurement model of the tracking filter defines the state space of atleast one detectable target, and a distribution function of said atleast one target at time taking into account all radar return signalsmeasurements made up to this time, wherein S_(θ) is discretized todiscrete intervals in a vertical-direction of a fixed horizontalcoordinate system, where B_(j) denotes these intervals, such thatS_(θ)=U_(j)B_(j).
 9. The method according to claim 8, wherein thediscretization is denser where the elevation distribution is high andless dense where the elevation distribution is small.
 10. The methodaccording to claim 5, wherein obtaining azimuth angle information of atleast one target radar return signal comprises: measuring targetparameters in said local coordinate system of said radar platform, andtransferring said target parameters to said tracking filter, which isconfigured to produce an estimate of the state at the current time stepbased on a state estimate from a previous time step.
 11. The methodaccording to claim 5, wherein estimating said azimuth angle of said atleast one target utilizing said tracking filter comprises: determiningcoordinate transfer functions g_(j) for all j, and transforming measuredtarget parameters in said local coordinate system to target parametersin the fixed reference coordinate system for all different Θ_(j)utilizing said coordinate transfer functions g_(j).
 12. The methodaccording to claim 5, wherein estimating said azimuth angle of said atleast one target utilizing said tracking filter further comprises:determining a likelihood function L of the measurement at time t thepresent state, and calculating the updated state estimate based upon thepredicted state estimate of the target motion, and the radar measurementinformation.